On the Shockley-Read-Hall Model:Generation-Recombination in SemiconductorsThierry Goudon1,Vera Miljanovi´c2,Christian Schmeiser3January17,2006AbstractThe Shockley-Read-Hall model for generation-recombination of electron-hole pairs insemiconductors based on a quasistationary approximation for electrons in a trappedstate is generalized to distributed trapped states in the forbidden band and to ki-netic transport models for electrons and holes.The quasistationary limit is rigorouslyjustified both for the drift-diffusion and for the kinetic model.Keywords:semiconductor,generation,recombination,drift-diffusion,kinetic modelAMS subject classification:Acknowledgment:This work has been supported by the European IHP network“HYKE-Hyperbolic and Kinetic Equations:Asymptotics,Numerics,Analysis”,contract no.HPRN-CT-2002-00282,and by the Austrian Science Fund,project no.W008(Wissenschaftskolleg ”Differential Equations”).Part of it has been carried out while the second and third authors enjoyed the hospitality of the Universit´e des Sciences et Technologies Lille1.1Team SIMPAF–INRIA Futurs&Labo.Paul Painlev´e UMR8524,CNRS–Universit´e des Sciences et Technologies Lille1,Cit´e Scientifique,F-59655Villeneuve d’Ascq cedex,France.2Wolfgang Pauli Institut,Universit¨a t Wien,Nordbergstraße15C,1090Wien,Austria.3Fakult¨a t f¨u r Mathematik,Universit¨a t Wien,Nordbergstraße15C,1090Wien,Austria&RICAM Linz,¨Osterreichische Akademie der Wissenschaften,Altenbergstr.56,4040Linz,Austria.11IntroductionThe Shockley-Read-Hall (SRH-)model was introduced in 1952[13],[9]to describe the sta-tistics of recombination and generation of holes and electrons in semiconductors occurring through the mechanism of trapping.The transfer of electrons from the valence band to the conduction band is referred to as the generation of electron-hole pairs (or pair-generation process),since not only a free electron is created in the conduction band,but also a hole in the valence band which can contribute to the charge current.The inverse process is termed recombination of electron-hole pairs.The bandgap between the upper edge of the valence band and the lower edge of the conduction band is very large in semiconductors,which means that a big amount of energy is needed for a direct band-to-band generation event.The presence of trap levels within the forbidden band caused by crystal impurities facilitates this process,since the jump can be split into two parts,each of them ’cheaper’in terms of energy.The basic mechanisms are illustrated in Figure 1:(a)hole emission (an electron jumps from the valence band to the trapped level),(b)hole capture (an electron moves from an occupied trap to the valence band,a hole disappears),(c)electron emission (an electron jumps from trapped level to the conduction band),(d)electron capture (an electron moves from the conduction band to an unoccupiedtrap).a b cd conduction band trapped level valence bandE v E ce n e r g y of a n e l e c t r o n ,E E Figure 1:The four basic processes of electron-hole recombination.Models for this process involve equations for the densities of electrons in the conduction band,holes in the valence band,and trapped electrons.Basic for the SRH model are the drift-diffusion assumption for the transport of electrons and holes,the assumption of one trap level in the forbidden band,and the assumption that the dynamics of the trapped electrons is quasistationary,which can be motivated by the smallness of the density of trapped states compared to typical carrier densities.This last assumption leads to the elimination of the density of trapped electrons from the system and to a nonlinear effective recombination-generation rate,reminiscent of Michaelis-Menten kinetics in chemistry.This model is an2important ingredient of simulation models for semiconductor devices(see,e.g.,[10],[12]).In this work,two generalizations of the classical SRH model are considered:Instead of a single trapped state,a distribution of trapped states across the forbidden band is allowed and,in a second step,a semiclassical kinetic model including the fermion nature of the charge carriers is introduced.Although direct band-to-band recombination-generation(see, e.g.,[11])and impact ionization(e.g.,[2],[3])have been modelled on the kinetic level before, this is(to the knowledge of the authors)thefirst attempt to derive a’kinetic SRH model’. (We mention also the modelling discussions and numerical simulations in??.) For both the drift-diffusion and the kinetic models with self consistent electricfields ex-istence results and rigorous results concerning the quasistationary limit are proven.For the drift-diffusion problem,the essential estimate is derived similarly to[6],where the quasi-neutral limit has been carried out.For the kinetic model Degond’s approach[4]for the existence of solutions of the Vlasov-Poisson problem is extended.Actually,the existence theory already provides the uniform estimates necessary for passing to the quasistationary limit.In the following section,the drift-diffusion based model is formulated and nondimension-alized,and the SRH-model is formally derived.Section3contains the rigorous justification of the passage to the quasistationary limit.Section4corresponds to Section2,dealing with the kinetic model,and in Section5existence of global solutions for the kinetic model is proven and the quasistationary limit is justified.2The drift-diffusion Shockley-Read-Hall modelWe consider a semiconductor crystal with a forbidden band represented by the energy interval (E v,E c)with the valence band edge E v and the conduction band edge E c.The constant(in space)number density of trap states N tr is obtained by summing up contributions across the forbidden band:N tr= E c E v M tr(E)dE.Here M tr(E)is the energy dependent density of available trapped states.The position density of occupied traps is given byn tr(f tr)(x,t)= E c E v M tr(E)f tr(x,E,t)dE,where f tr(x,E,t)is the fraction of occupied trapped states at position x∈Ω,energy E∈(E v,E c),and time t≥0.Note that0≤f tr≤1should hold from a physical point of view.The evolution of f tr is coupled to those of the density of electrons in the conduction band, denoted by n(x,t)≥0,and the density of holes in the valence band,denoted by p(x,t)≥0. Electrons and holes are oppositely charged.The coupling is expressed through the following3quantitiesS n=1τn N tr n0f tr−n(1−f tr) ,S p=1τp N tr p0(1−f tr)−pf tr ,(1)R n= E c E v S n M tr dE,R p= E c E v S p M tr dE.(2) Indeed,the governing equations are given by∂t f tr=S p−S n=p0τp N tr+nτn N tr−f tr p0+pτp N tr+n0+nτn N tr ,(3)∂t n=∇·J n+R n,J n=µn(U T∇n−n∇V),(4)∂t p=−∇·J p+R p,J p=−µp(U T∇p+p∇V),(5)εs∆V=q(n+n tr(f tr)−p−C).(6) For the current densities J n,J p we use the simplest possible model,the drift diffusion ansatz,with constant mobilitiesµn,µp,and with thermal voltage U T.Moreover,since the trapped states havefixed positions,noflux appears in(3).By R n and R p we denote the recombination-generation rates for n and p,respectively. The rate constants areτn(E),τp(E),n0(E),p0(E),where n0(E)p0(E)=n i2with the energy independent intrinsic density n i.Integration of(3)yields∂t n tr=R p−R n.(7) By adding equations(4),(5),(7),we obtain the continuity equation∂t(p−n−n tr)+∇·(J n+J p)=0,(8) with the total charge density p−n−n tr and the total current density J n+J p.In the Poisson equation(6),V(x,t)is the electrostatic potential,εs the permittivity of the semiconductor material,q the elementary charge,and C=C(x)the given doping profile.Note that ifτn,τp,n0,p0are independent from E,or if there exists only one trap level E trwith M tr(E)=N trδ(E−E tr),then R n=1τn [n0n trN tr−n(1−n tr N tr)],R p=1τp[p0(1−n tr N tr)−p n tr N tr],and the equations(4),(5)together with(7)are a closed system governing the evolution of n,p,and n tr.We now introduce a scaling of n,p,and f tr in order to render the equations(4)-(6) dimensionless:Scaling of parameters:i.M tr→N tr E c−E v M tr.ii.τn,p→¯ττn,p,where¯τis a typical value forτn andτp.iii.µn,p→¯µµn,p,where¯µis a typical value forµn,p.4iv.(n 0,p 0,n i ,C )→¯C(n 0,p 0,n i ,C ),where ¯C is a typical value of C .Scaling of unknowns:v.(n,p )→¯C(n,p ).vi.n tr →N tr n tr .vii.V →U T V .viii.f tr →f tr .Scaling of independent variables:ix.E →E v +(E c −E v )E .x.x →√¯µU T ¯τx ,where the reference length is a typical diffusion length before recombina-tion.xi.t →¯τt ,where the reference time is a typical carrier life time.Dimensionless parameters:xii.λ= εs q ¯C ¯µ¯τ=1¯x εs U T q ¯C is the scaled Debye length.xiii.ε=N tr ¯C is the ratio of the density of traps to the typical doping density,and will be assumed to be small:ε≪1.The scaled system reads:ε∂t f tr =S p (p,f tr )−S n (n,f tr ),S p =1τpp 0(1−f tr )−pf tr ,S n =1τn n 0f tr −n (1−f tr ) ,(9)∂t n =∇·J n +R n (n,f tr ),J n =µn (∇n −n ∇V ),R n = 10S n M tr dE ,(10)∂t p =−∇·J p +R p (p,f tr ),J p =−µp (∇p +p ∇V ),R p = 10S p M tr dE ,(11)λ2∆V =n +εn tr −p −C ,n tr (f tr )= 10f tr M tr dE ,(12)with n 0(E )p 0(E )=n 2i and 10M tr dE =1.5By lettingε→0in(9)formally,we obtain f tr=τn p0+τp nτn(p+p0)+τp(n+n0),and the reduced systemhas the following form∂t n=∇·J n+R(n,p),(13)∂t p=−∇·J p+R(n,p),(14)R(n,p)=(n i2−np) 10M tr(E)τn(E)(p+p0(E))+τp(E)(n+n0(E))dE,(15)λ2∆V=n−p−C.(16) Note that ifτn,τp,n0,p0are independent from E or if there exists only one trap level,thenwe would have the standard Shockley-Read-Hall model,with R=n i2−npτn(p+p0)+τp(n+n0).Existenceand uniqueness of solutions of the limiting system(13)–(16)under the assumptions(21)–(25) stated below is a standard result in semiconductor modelling.A proof can be found in,e.g., [10].3Rigorous derivation of the drift-diffusion Shockley-Read-Hall modelWe consider the system(9)–(12)with the position x varying in a bounded domainΩ∈R3 (all our results are easily extended to the one-and two-dimensional situations),the energy E∈(0,1),and time t>0,subject to initial conditionsn(x,0)=n I(x),p(x,0)=p I(x),f tr(x,E,0)=f tr,I(x,E)(17) and mixed Dirichlet-Neumann boundary conditionsn(x,t)=n D(x,t),p(x,t)=p D(x,t),V(x,t)=V D(x,t)x∈∂ΩD⊂∂Ω(18) and∂n ∂ν(x,t)=∂p∂ν(x,t)=∂V∂ν(x,t)=0x∈∂ΩN:=∂Ω\∂ΩD,(19)whereνis the unit outward normal vector along∂ΩN.We permit the special cases thateither∂ΩD or∂ΩN are empty.More precisely,we assume that either∂ΩD has positive(d−1)-dimensional measure,or it is empty.In the second situation(∂ΩD empty)we haveto assume total charge neutrality,i.e.,Ω(n+εn tr−p−C)dx=0,if∂Ω=∂ΩN.(20)The potential is then only determined up to a(physically irrelevant)additive constant.The following assumptions on the data will be used:For the boundary datan D,p D∈W1,∞loc(Ω×R+t),V D∈L∞loc(R+t,W1,6(Ω)),(21)6for the initial datan I ,p I ∈H 1(Ω)∩L ∞(Ω),0≤f tr,I ≤1,(22) Ω(n I +εn tr (f tr,I )−p I −C )dx =0,if ∂Ω=∂ΩN ,(23)for the doping profile C ∈L ∞(Ω),(24)for the recombination-generation rate constants n 0,p 0,τn ,τp ∈L ∞((0,1)),τn ,τp ≥τmin >0.(25)With these assumptions,a local existence and uniqueness result for the problem (9)–(12),(17)–(19)for fixed positive εcan be proven by a straightforward extension of the approach in[5](see also [10]).In the following,local existence will be assumed,and we shall concentrate on obtaining bounds which guarantee global existence and which are uniform in εas ε→0.For the sake of simplicity,we consider that the data in (21),(22)and (24)do not depend on ε;of course,our strategy works dealing with sequences of data bounded in the mentioned spaces.The following result is a generalization of [6,Lemma 3.1],where the case of homogeneous Neumann boundary conditions and vanishing recombination was treated.Our proof uses a similar approach.Lemma 3.1.Let the assumptions (21)–(25)be satisfied.Then,the solution of (9)–(12),(17)–(19)exists for all times and satisfies n,p ∈L ∞loc ((0,∞),L ∞(Ω))∩L 2loc ((0,∞),H 1(Ω)))uniformly in εas ε→0as well as 0≤f tr ≤1.Proof.Global existence will be a consequence of the following estimates.Introducing the new variables n =n −n D , p =p −p D , C=C −εn tr −n D +p D the equations (10)–(12)take the following form:∂t n =∇·J n +R n −∂t n D ,J n =µn ∇ n +∇n D −( n +n D )∇V ,(26)∂t p =−∇J p +R p −∂t p D ,J p =−µp ∇ p +∇p D +( p +p D )∇V ,(27)λ2∆V = n − p − C.(28)As a consequence of 0≤f tr ≤1, C∈L ∞((0,∞)×Ω)holds.For q ≥2and even,we multiply (26)by n q −1/µn ,(27)by p q −1/µp ,and add:d dt Ω n q qµn + p q qµp dx =−(q −1) Ω n q −2∇ n ∇n dx −(q −1) Ω p q −2∇ p ∇p dx +(q −1) Ω n q −2n ∇ n − p q −2p ∇ p ∇V dx + Ω n q −1µn (R n −∂t n D )+ Ω p q −1µp (R p −∂t p D )=:I 1+I 2+I 3+I 4+I 5.(29)7Using the assumptions on n D ,p D and |R n |≤C (n +1),|R p |≤C (p +1),we estimate I 4≤C Ω| n |q −1(n +1)dx ≤C Ω n q dx +1 ,I 5≤C Ωp q dx +1 .The term I 3can be rewritten as follows:I 3= Ω n q −1∇ n − p q −1∇ p ]∇V dx + Ω n q −2∇ n (n D ∇V )dx − Ω p q −2∇ p (p D ∇V )dx =−1λ2q Ω[ n q − p q ]( n − p − C )dx −1λ2(q −1) Ω n q −1(∇n D ∇V +n D ( n − p − C ) dx +1λ2(q −1) Ωp q −1(∇p D ∇V +p D ( n − p − C ) dx.The second equality uses integration by parts and (28).The first term on the right hand side is the only term of degree q +1.It reflects the quadratic nonlinearity of the problem.Fortunately,it can be written as the sum of a term of degree q and a nonnegative term.By estimation of the terms of degree q using the assumptions on n D and p D as well as ∇V L q (Ω)≤C ( n L q (Ω)+ p L q (Ω)+ CL q (Ω)),we obtain I 3≤−1λ2q Ω[ n q − p q ]( n − p )dx +C Ω( n q + p q )dx +1 ≤C Ω( n q + p q )dx +1 .The integral I 1can be written as I 1=− Ω n q −2|∇n |2dx + Ω n q −2∇n D ∇n dx.(30)By rewriting the integrand in the second integral asn q −2∇n D ∇n = n q −22∇n n q −22∇n Dand applying the Cauchy-Schwarz inequality,we have the following estimate for (30):I 1≤− Ωn q −2|∇n |2dx +Ω n q −2|∇n |2dx Ω n q −2|∇n D |2dx ≤−12 Ω n q −2|∇n |2dx +C n q −2L q ≤−12 Ωn q −2|∇n |2dx +C Ω n q dx +1 .8For I2,the same reasoning(with n and n D replaced by p and p D,respectively)yields an analogous estimate.Collecting our results,we obtaind dt Ω n q qµn+ p q qµp dx≤−12 Ω n q−2|∇n|2dx−12 Ω p q−2|∇p|2dx+C Ω( n q+ p q)dx+1 .(31)Since q≥2is even,thefirst two terms on the right hand side are nonpositive and the Gronwall lemma givesΩ( n q+ p q)dx≤e qCt Ω( n(t=0)q+ p(t=0)q)dx+1 .A uniform-in-q-and-εestimate for n L q, p L q follows,and the uniform-in-εbound in L∞loc((0,∞),L∞(Ω))is obtained in the limit q→∞.The estimate in L2loc((0,∞),H1(Ω)) is then derived by returning to(31)with q=2.Now we are ready for proving the main result of this section.Theorem3.2.Let the assumptions of Theorem3.1be satisfied.Then,asε→0,for every T>0,the solution(f tr,n,p,V)of(9)–(12),(17)–(19)converges with convergence of f tr in L∞((0,T)×Ω×(0,1))weak*,n and p in L2((0,T)×Ω),and V in L2((0,T),H1(Ω)).The limits of n,p,and V satisfy(13)–(19).Proof.The L∞-bounds for f tr,n,and p,and the Poisson equation(12)imply∇V∈L2((0,T)×Ω).From the definition of J n,J p(see(4),(5)),it then follows that J n,J p∈L2((0,T)×Ω).Then(10)and(11)together with R n,R p∈L∞((0,T)×Ω)imply∂t n,∂t p∈L2((0,T),H−1(Ω)).The previous result and the Aubin lemma(see,e.g.,Simon[14,Corollary 4,p.85])gives compactness of n and p in L2((0,T)×Ω).We already know from the Poisson equation that∇V∈L∞((0,T),H1(Ω)).By taking the time derivative of(12),one obtains∂t∆V=∇·(J n+J p),with the consequence that∂t∇V is bounded in L2((0,T)×Ω).Therefore,the Aubin lemma can again be applied as above to prove compactness of∇V in L2((0,T)×Ω).These results and the weak compactness of f tr are sufficient for passing to the limit in the nonlinear terms n∇V,p∇V,nf tr,and pf tr.Let us also remark that∂t n and∂t p are bounded in L2(0,T;H−1(Ω)),so that n,p are compact in C0([0,T];L2(Ω)−weak).With this remark the initial data for the limit equation makes sense.By the uniqueness result for the limiting problem(mentioned at the end of Section2),the convergence is not restricted to subsequences.94A kinetic Shockley-Read-Hall modelIn this section we replace the drift-diffusion model for electrons and holes by a semiclassical kinetic transport model.It is governed by the system∂t f n+v n(k)·∇x f n+q∇x V·∇k f n=Q n(f n)+Q n,r(f n,f tr),(32)∂t f p+v p(k)·∇x f p−q ∇x V·∇k f p=Q p(f p)+Q p,r(f p,f tr),(33)∂t f tr=S p(f p,f tr)−S n(f n,f tr),(34)εs∆x V=q(n+n tr−p−C),(35) where f i(x,k,t)represents the particle distribution function(with i=n for electrons and i=p for holes)at time t≥0,at the position x∈R3,and at the wave vector(or generalized momentum)k∈R3.All functions of k have the periodicity of the reciprocal lattice of the semiconductor crystal.Equivalently,we shall consider only k∈B,where B is the Brillouin zone,i.e.,the set of all k which are closer to the origin than to any other lattice point,with periodic boundary conditions on∂B.The coefficient functions v n(k)and v p(k)denote the electron and hole velocities,respec-tively,which are related to the electron and hole band diagrams byv n(k)=∇kεn(k)/ ,v p(k)=−∇kεp(k)/ ,where is the reduced Planck constant.The elementary charge is still denoted by q.The collision operators Q n and Q p describe the interactions between the particles and the crystal lattice.They involve several physical phenomena and can be written in the general formQ n(f n)= B Φn(k,k′)[M n f′n(1−f n)−M′n f n(1−f′n)]dk′,(36)Q p(f p)= B Φp(k,k′)[M p f′p(1−f p)−M′p f p(1−f′p)]dk′,(37) with the primes denoting evaluation at k′,with the nonnegative,symmetric scattering cross sections Φn(k,k′)and Φp(k,k′),and with the MaxwelliansM n(k)=c n exp(−εn(k)/k B T),M p(k)=c p exp(−εp(k)/k B T),where k B T is the thermal energy of the semiconductor crystal lattice and the constants c n, c p are chosen such that B M n dk= B M p dk=1.The remaining collision operators Q n,r(f n,f tr)and Q p,r(f p,f tr)model the generation and recombination processes and are given byQ n,r(f n,f tr)= E c E vˆS n(f n,f tr)M tr dE,(38)10withˆS n (f n ,f tr )=Φn (k,E )N tr[n 0M n f tr (1−f n )−f n (1−f tr )],and Q p,r (f p ,f tr )= E c E vˆS p (f p ,f tr )M tr dE ,(39)with ˆS p (f p ,f tr )=Φp (k,E )N tr[p 0M p (1−f p )(1−f tr )−f p f tr ],and where Φn,p are non negative and M tr (x,E )is the density of available trapped states as for the drift diffusion model,except that we allow for a position dependence now.This will be commented on below.The parameter N tr is now determined as N tr =sup x ∈R 3 10M tr (x,E )dE .The right hand side in the equation for the occupancy f tr (x,E,t )of the trapped states is defined byS n (f n ,f tr )= BˆS n dk =λn [n 0M n (1−f n )]f tr −λn [f n ](1−f tr ),(40)with λn [g ]= B Φn g dk ,andS p (f p ,f tr )=BˆS p dk =λp [p 0M p (1−f p )](1−f tr )−λp [f p ]f tr ,(41)with λp [g ]= B Φp g dk .The factors (1−f n )and (1−f p )take into account the Pauli exclusion principle,which therefore manifests itself in the requirement that the values of the distribution function have to respect the bounds 0≤f n ,f p ≤1.The position densities on the right hand side of the Poisson equation (35)are given byn (x,t )= B f n dk ,p (x,t )= B f p dk ,n tr (x,t )=E cE v f tr M tr dE.The following scaling,which is strongly related to the one used for the drift-diffusion model,will render the equations (32)-(35)dimensionless:Scaling of parameters:i.M tr →N tr E v −E cM tr ,ii.(εn ,εp )→k B T (εn ,εp ),with the thermal energy k B T ,iii.(Φn ,Φp )→τ−1rg (Φn ,Φp ),where τrg is a typical carrier life time,iv.( Φn , Φp )→τ−1coll( Φn , Φp ),v.(n 0,p 0,C )→C (n 0,p 0,C ),where C is a typical value of |C |,11vi.(M n,M p)→C−1(M n,M p).Scaling of independent variables:vii.x→k B T√τrgτcoll C−1/3 −1x,viii.t→τrg t,ix.k→C1/3k,x.E→E v+(E c−E v)E,Scaling of unknowns:xi.(f n,f p,f tr)→(f n,f p,f tr),xii.V→U T V,with the thermal voltage U T=k B T/q.Dimensionless parameters:xiii.α2=τcoll,τrgxiv.λ=q√τrgτcoll C1/6 εs k B T,,where again we shall study the situationε≪1.xv.ε=N trCFinally,the scaled system readsα2∂t f n+αv n(k)·∇x f n+α∇x V·∇k f n=Q n(f n)+α2Q n,r(f n,f tr),(42)α2∂t f p+αv p(k)·∇x f p−α∇x V·∇k f p=Q p(f p)+α2Q p,r(f p,f tr),(43)ε∂t f tr=S p(f p,f tr)−S n(f n,f tr),(44)λ2∆x V=n+εn tr−p−C=−ρ,(45) with v n=∇kεn,v p=−∇kεp,with Q n and Q p still having the form(36)and,respectively, (37),with the scaled MaxwelliansM n(k)=c n exp(−εn(k)),M p(k)=c p exp(−εp(k)),(46) and with the recombination-generation termsQ n,r(f n,f tr)= 10ˆS n M tr dE,Q p,r(f p,f tr)= 10ˆS p M tr dE,(47) withˆS=Φn[n0M n f tr(1−f n)−f n(1−f tr)],ˆS p=Φp[p0M p(1−f tr)(1−f p)−f p f tr].(48) n12The right hand side of (44)still has the form (40),(41).The position densities are given byn = B f n dk ,p = B f p dk ,n tr = 1f tr M tr dE.(49)The system (42)–(44)conserves the total charge ρ=p +C −n −εn tr .With the definitionJ n =−1α B v n f n dk ,J p =1α Bv p f p dk ,of the current densities,the following continuity equation holds formally:∂t ρ+∇x ·(J n +J p )=0.Setting formally ε=0in (44)we obtainf tr (f n ,f p )=p 0λp [M p (1−f p )]+λn [f n ]p 0λp [M p (1−f p )]+λp [f p ]+λn [f n ]+n 0λn [M n (1−f n )]Substitution f tr into (47)leads to the kinetic Shockley-Read-Hall recombination-generation operatorsQ n,r (f n ,f p )=g n [f n ,f p ](1−f n )−r n [f n ,f p ]f n ,Q p,r (f n ,f p )=g p [f n ,f p ](1−f p )−r p [f n ,f p ]f p ,(50)withg n =10Φn M n n 0 p 0λp [M p (1−f p )]+λn [f n ] M tr p 0λp [M p (1−f p )]+λp [f p ]+λn [f n ]+n 0λn [M n (1−f n )]dE ,r n =10Φn λp [f p ]+n 0λn [M n (1−f n )] M tr p 0λp [M p (1−f p )]+λp [f p ]+λn [f n ]+n 0λn [M n (1−f n )]dE ,g p =10Φp M p p 0 n 0λn [M n (1−f n )]+λp [f p ] M tr p 0λp [M p (1−f p )]+λp [f p ]+λn [f n ]+n 0λn [M n (1−f n )]dE ,r p = 1Φp λn [f n ]+p 0λp [M p (1−f p )] M trp 0λp [M p (1−f p )]+λp [f p ]+λn [f n ]+n 0λn [M n (1−f n )]dE.Of course,the limiting model still conserves charge,which is expressed by the identityB Q n,r dk = BQ p,r dk.Pairs of electrons and holes are generated or recombine,however,in general not with the same wave vector.This absence of momentum conservation is reasonable since the process involves an interaction with the trapped states fixed within the crystal lattice.135Rigorous derivation of the kinetic Shockley-Read-Hall modelThe limitε→0will be carried out rigorously in an initial value problem for the kineticmodel with x∈R3.Concerning the behaviour for|x|→∞,we shall require the densities tobe in L1and use the Newtonian potential solution of the Poisson equation,i.e.,(45)will bereplaced byE(x,t)=−∇x V=λ−2 R3x−y|x−y|3ρ(y,t)dy.(51) We define Problem(K)as the system(42)–(44),(51)with(36),(37),(47)–(49),(40),and(41),subject to the initial conditionsf n(x,k,0)=f n,I(x,k),f p(x,k,0)=f p,I(x,k),f tr(x,E,0)=f tr,I(x,E).We start by stating our assumptions on the data.For the velocities we assumev n,v p∈W1,∞per(B),(52) where here and in the following,the subscript per denotes Sobolev spaces of functions of ksatisfying periodic boundary conditions on∂B.Further we assume that the cross sectionssatisfyΦn, Φp≥0, Φn, Φp∈W1,∞per(B×B),(53) andΦn,Φp≥0,Φn,Φn∈W1,∞per(B×(0,1)).(54) Afinite total number of trapped states is assumed:M tr≥0,M tr∈W1,∞(R3×(0,1))∩W1,1(R3×(0,1)).The L1-assumption with respect to x is needed for controlling the total number of generatedparticles.For the initial data we assume0≤f n,I,f p,I≤1,f n,I,f p,I∈W1,∞per(R3×B)∩W1,1per(R3×B),0≤f tr,I≤1,f tr,I∈W1,∞per(R3×(0,1)).(55) We also assumen0,p0∈L∞((0,1)),C∈W1,∞(R3)∩W1,1(R3).(56) Finally,we need an upper bound for the life time of trapped electrons:B(Φn min{1,n0M n}+Φp min{1,p0M p})dk≥γ>0.(57)The reason for the various differentiability assumptions above is that we shall constructsmooth solutions by an approach along the lines of[11],which goes back to[4].14An essential tool are the following potential theory estimates[15]:E L∞(R3)≤C ρ 1/3L1(R3) ρ 2/3L∞(R3),(58)∇x E L∞(R3)≤C 1+ ρ L1(R3)+ ρ L∞(R3) 1+log(1+ ∇xρ L∞(R3)) .(59) We start by rewriting the collision and recombination generation operators asQ i(f i)=a i[f i](1−f i)−b i[f i]f i,i=n,p,andQ i,r(f i,f tr)=g i[f tr](1−f i)−r i[f tr]f i,i=n,p,witha i[f i]= B Φi M i f′i dk′,b i[f i]= B Φi M′i(1−f′i)dk′,i=n,pg n[f tr]= 10Φn n0M n f tr M tr dE,g p[f tr]= 10Φp p0M p(1−f tr)M tr dE,r n[f tr]= 10Φn(1−f tr)M tr dE,r p[f tr]= 10Φp f tr M tr dE.In order to construct an approximating sequence(f j n,f j p,f j tr,E j)we begin withf0i(x,k,t)=f i,I(x,k),i=n,p,f0tr(x,E,t)=f tr,I(x,E)(60) Thefield always satisfiesE j(x,t)= R3x−y|x−y|3ρj(y,t)dy(61)Let(f j n,f j p,f j tr,E j)be given.Then the f i j+1are defined as the solutions of the following problem:α2∂t f j+1n +αv n(k)·∇x f j+1n−αE j·∇k f j+1n=(a n[f j n]+α2g n[f j tr])(1−f j+1n)−(b n[f j n]+α2r n[f j tr])f j+1n,α2∂t f j+1p +αv p(k)·∇x f j+1p+αE j·∇k f j+1p=(a p[f j p]+α2g p[f j tr])(1−f j+1p)−(b p[f j p]+α2r p[f j tr])f j+1p,ε∂t f j+1tr =(p0λp[M p(1−f j p)]+λn[f j n])(1−f j+1tr)−(n0λn[M n(1−f j n)]+λp[f j p])f j+1tr,(62)subject to the initial conditionsf j+1 n (x,k,0)=f n,I(x,k),f j+1p(x,k,0)=f p,I(x,k),f j+1tr(x,E,0)=f tr,I(x,E).(63)For the iterative sequence we state the following lemma,which is very similar to the Propo-sition3.1from[11]:15Lemma5.1.Let the assumptions(52)–(56)be satisfied.Then the sequence(f j n,f j p,f j tr,E j), defined by(60)-(63)satisfies for any time T>0a)0≤f i j≤1,i=n,p,tr.b)f j n and f j p are uniformly bounded with respect to j→∞andε→0in L∞((0,T),L1(R3×B).c)E j is uniformly bounded with respect to j andεin L∞((0,T)×R3).Proof.Thefirst two equations in(62)are standard linear transport equations,and the third equation is a linear ODE.Existence and uniqueness for the initial value problems are therefore standard results.Note that the a i,b i,g i,r i,andλi in(62)are nonnegative if we assume that a)holds for j.Then a)for j+1is an immediate consequence of the maximum principle.To estimate the L1-norms of the distributions,we integrate thefirst equation in(62)and obtainf j+1n L1(R3×B)≤ f n,I L1(R3×B)+ t0 a n[f j n]1α2+g n[f j tr] L1(R3×B)(s)ds.(64) The boundedness of Φn,Φn,and f j tr,and the integrability of M tr implya n[f j n] L1(R3×B)≤C f j n L1(R3×B), g n[f j tr] L1(R3×B)≤C.(65) Now this is used in(64).Then an estimate is derived for f j n by replacing j+1by j and using the Gronwall inequality.Finally,it is easily since that this estimate is passed from j to j+1by(64).An analogous argument for f j p completes the proof of b).A uniform-in-ε(L1∩L∞)-bound for the total charge densityρj=n j+εn j tr−p j−C follows from b)and from the integrability of M tr.The statement c)of the lemma is now a consequence of(58).For passing to the limit in the nonlinear terms some compactness is needed.Therefore we prove uniform smoothness of the approximating sequence.Lemma5.2.Let the assumptions(52)–(57)be satisfied.Then for any time T>0:a)f j n and f j p are uniformly bounded with respect to j andεin L∞((0,T),W1,1per(R3×B)∩(R3×B)),W1,∞perb)f j tr is uniformly bounded with respect to j andεin L∞((0,T),W1,∞(R3×(0,1))),c)E j is uniformly bounded with respect to j andεin L∞((0,T),W1,∞(R3)).16。
